Commun. Math. Phys. 164, 473-488 (1994)
Communications IΠ
Mathematical Physics
© Springer-Verlag 1994
Classical Intertwiner Space and Quantization Ping Xu* Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720, USA, and Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA. email:
[email protected] Received: 15 April 1993/in revised form: 15 November 1993
Abstract: Given two symplectic realizations, a symplectic manifold called the classical intertwiner space is introduced as a classical analogue of an intertwiner space of representations of an associative algebra. We describe explicitly how a quantum data on realizations induces a quantum data on their classical intertwiner space. 1. Introduction Let G be a compact Lie group and X a Hamiltonian G-space. G is thus considered as a symmetry group of the classical space X. According to the "creed" of geometric quantization, if the classical space X is quantized to a quantum phase space (i.e., a Hubert space), G becomes a symmetry group of the corresponding quantum space. This idea has in fact inspired many significant results in mathematical physics, among them orbit method of group representations and theory of geometric quantization [9,19]. The intertwiner space between two representations p1: G -> E n d ^ ) and ρ2: G->End(V 2 ) is, by definition, HomG(V2? Vi\ the space of all G-equivariant linear maps from V2 to Vι. When pγ is irreducible, the dimension of HomG(V2? V\) is usually called the multiplicity of px in p2, a fundamental concept in representation theory. The classical counterpart of Hom G (y 2 ? VΊ) is closely related to the so-called Marsden-Weinstein reduction [16]. In fact, the classical intertwiner space, for any symplectic homogeneous spaces X1 and X2, is defined as the symplectic (or Marsden-Weinstein) reduced space (X2 x Xι)0. A remarkable result of Guillemin-Sternberg [5] asserts that for a Kahler manifold, the geometric quantization of classical intertwiner space is isomorphic to the intertwiner space of the corresponding representations.
* Research partially supported by NSF Grant DMS92-03398 and at MSRI supported by NSF Grant DMS90-22140
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Recently, more and more interest has been drawn to the so-called hidden symmetry due to the rapid development of quantum group theory. In this case, on the quantum level, the quantum symmetry is represented by a representation of a quantum group, while the classical symmetry is played by a Poisson group homogeneous space. It is therefore desirable to extend Guillemin-Sternberg theory to this more general setting. To approach this, we shall put this problem into a slightly more general framework. Recall that a Hamiltonian G-space X corresponds to an equivariant momentum map J: X -• g*. J is in fact a Poisson map if g* is equipped with the natural Lie-Poisson structure. Moreover, such a Poisson map uniquely determines the G-action on X. Such a pair (X, J) is also called a symplectic realization of the Lie-Poisson space g*. For a general Poisson manifold P, a symplectic realization (X, J) is defined as a symplectic manifold X together with a Poisson map J : X-*P which is complete in the sense that the pullback to X of every compactly supported function on P has a complete Hamiltonian vector field1. As described early in this introduction, a symplectic realization of a Lie-Poisson space g* is the classical analogue of a representation of the group G or, what is nearly equivalent, of the universal enveloping algebra ί/(g). Since £/(g) may be thought of as a quantum deformation of the Poisson algebra of (polynomial) functions on g*, it is a natural generalization of this situation to consider realizations of a general Poisson manifold P as the classical analogues of representations of a quantum deformation of the functions on P. The following table should be useful for understanding the relationship between various notions in this general setting: CLASSICAL symplectic groupoids of Poisson manifolds
QUANTUM -•
associative algebras
symplectic realizations of Poisson manifolds -• representations of associative algebras classical intertwiner space
-»
intertwiner of representations
The horizontal arrows represent processes generally known as "quantization." Our purpose is to extend as much as possible the Guillemin-Sternberg theory to this general non-linear framework. This paper consists of our first attempt toward this direction. To any two symplectic realizations J1: X1-^ P and J2: X2 -* P, there associates in general a symplectic manifold which is a certain quotient space of (X2 *Xi). This symplectic manifold is called the classical intertwiner space. Here, X2*X\ denotes the inverse image of the diagonal in P x P under the map Jί x J2, and the bar means reversing the sign of the symplectic structure. This classical intertwiner space is closely related to the symplectic groupoid Γ of P, and in fact is the quotient space Γ\(X2 *X\) under the diagonal Γ-action (see Sect. 4 for the precise definition). When P is the Lie-Poisson g* (or even dual of a general Poisson group G*), this space reduces to the usual intertwiner space [5]. In this paper, we shall describe explicitly how a quantum data on the symplectic groupoid and those on the realizations XUX2 induce a quantum data on the classical intertwiner space 1
Note that in some previous literature [2, 26, 28], they are called complete symplectic realizations while the terminology of symplectic realizations refers to Poisson maps J: X-+P without any assumption on completeness.
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Γ\(X2 *Xi) Here, as usual, by a quantum data on a symplectic manifold, we mean the following two structures: prequantum circle-bundle and polarization. The paper is organized as follows. In Sect. 2, we present some basic facts regarding central extensions of Lie algebroids. Section 3 concerns central extensions of symplectic groupoids. Section 4 introduces symplectic realizations and classical intertwiner spaces. Section 5 is devoted to the discussion on prequantization of symplectic realizations. In particular, we prove that the symplectic groupoid central extension described in Sect. 3 naturally acts on the prequantum bundle of any realization. This result will be needed for the construction in Sect. 6. Section 6 contains the main result of this paper, where an explicit construction for the prequantum bundle of a classical intertwiner space is given. Section 7 is devoted to the discussion on the aspect of polarizations. Finally, some discussions are outlined in Sect. 8. We would like to thank Jiang-hua Lu and Alan Weinstein for their helpful advice in the course of this work.
2. Central Extensions of a Lie Algebroid
The notion of Lie algebroids is a natural generalization of that of Lie algebras. In fact, there are many notions and theorems for Lie algebroids which are parallel to those for Lie algebras. Let us first recall the definition of a Lie algebroid. Definition 2.1. A Lie algebroid over a manifold P is a vector bundle π: A -> P together with a vector bundle map p: A —• TP, called the anchor of A, and a Lie algebra structure on Γ(A), the space of sections of A, satisfying 2. lXjγ
]=flX9Y]+p{X)(f)Y9X9YeΓ{A),
Let A -• P be a Lie algebroid with anchor p. Suppose that ξ is a Lie algebroid 2-cocycle [14], i.e., a section of the vector bundle Λ 2 A * satisfying the condition that
-ξ{[Y9ZlX)
= 09
where A* is the dual bundle of A Let i = . 4 © ( P x ]R), the vector bundle direct sum of A with the trivial vector bundle P x R -> P. It is clear that any section of A can be identified with a pair (X,f), where X is a section of A a n d / a function on P. Define p:Ά-+TP by p(a9t) = p(a)9 for any aeA and ί e R . Also, we define the bracket of sections of A by
for any X, YeΓ(A\ and /, geCςχ>(P). It is easy to check that A becomes a Lie algebroid, which we shall call the central extension of the Lie algebroid A by the cocycle ξ. In particular, if ξ = 09 then the bracket of A is defined by Ϊp(X)g-p(Y)f).
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Such an extension is called the trivial extension of A by R. In this case, the natural embedding i: A-> A given by i(a) = (a, 0) is clearly a Lie algebroid morphism in the sense of [7] (see [15] also). The proof of the following proposition is straightforward. Proposition 2.1. If the 2-cocycle ξ is a coboundary ξ = dσ for some σeΓ(A*\ ξ(X9 Y) = p(X)σ(Y)-p(Y)σ(X)-σ([X,
i.e.,
F]),
the central extension by ξ is then isomorphic to the trivial central extension under the map: φ: A->A(B(PxlR), φ(a, t) = (a, t + σ(a)), where aeA, ίelR. Remark. It is well-known that the dual bundle of a Lie algebroid naturally carries a Poisson structure which is called Lie-Poisson structure [2]. The Lie-Poisson structure of a central extension of a Lie algebra is closely related to affine Poisson structures. Therefore it should be quite interesting to investigate in detail what kind of Poisson structures the dual of a central extension of a Lie algebroid corresponds to. Below is the very example in which we are most interested in this paper. Let P be a Poisson manifold with Poisson tensor π. The cotangent bundle T*P -+ P carries a natural Lie algebroid structure [2], which can be described as follows. Given fe Coo(P), denote the Hamiltonian vector field corresponding to / by Xf. Then the anchor p = π%\ T*P-+TP is determined by p(fdg)=fXg. Given ω,θeΩ1(P)9 and writing Xω = pω and Xθ = ρθ, the Lie algebroid bracket is
{ω,θ}=LXωθ-LXθω-d[π(ω,θ)
] .
(1)
The Poisson tensor π can be considered as a Lie algebroid 2-cocycle in a natural way [8,24]. Therefore, there associates a canonical central extension Ά = Γ * P ® ( P x I R ) . The global object corresponding to A is closely related to the prequantization of the symplectic groupoid, which is to be described in the following section. Remark. Lu has observed that this 2-cocycle is a coboundary iff there is a vector field V on P such that π = [V, π] =Lvπ. In other words, V is a Liouville vector field of the Poisson manifold P. Thus according to Proposition 2.1, the existence of a Liouville vector field is equivalent to saying that A is isomorphic to the trivial central extension. It is easily seen that any Lie-Poisson space g* possesses such a property. 3. Symplectic Groupoids and Prequantization The global objects corresponding to Lie algebroids are the so-called Lie groupoids. Briefly, a groupoid is a small category whose morphisms are all invertible. Below is the precise definition. We refer the reader to [14] for more details on Lie groupoid theory. Definition 3.1. A Lie groupoid Γ over a manifold P is a pair (Γ, P) equipped with surjective submersions α, β: Γ -> P (called the source and target maps respectively); a multiplication map m from Γ2 = {(g,h)eΓ x Γ\β(g) = a(h)} to Γ, an injection ε: P -> Γ (called the unit map); and an inverse mapping i: Γ -• Γ satisfying (where m(g,h) is denoted by gh, and i(g) by g~x):
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1. (associativity) g(hk) = (gh)k, which means that if one side is defined, so is the other and they are equal; 2. (identities) ε(oc(g))g = gε(β(g)); _ 1 1 3. (inverses) gg = ε(a(g)) and g g = ε(β(g)). We usually use the notation (ΓzJ P, α, β) to denote such a Lie groupoid. Associated to any Lie groupoid, there is a Lie algebroid as an infinitesimal invariant. However, conversely, a Lie algebroid may not always be the Lie algebroid of a Lie groupoid. In the case that it does arise from a Lie groupoid, one says that the Lie algebroid is integrable [14]. In particular, a Lie groupoid corresponding to the Lie algebroid arising from a Poisson manifold as described by Eq. (1) usually carries a symplectic structure, which is compatible with the groupoid structure in a certain sense. Such a groupoid is called a symplectic groupoid. More precisely, a symplectic groupoid is a Lie groupoid (Γ z | P, α, β) with a symplectic structure on Γ such that the graph oί_ multiplication A = {(g, h, gh)\β(g) = ot(h)} is a lagrangian submanifold of Γ x Γ x Γ (the bar denotes reversing the sign of the symplectic structure). For more details on symplectic groupoids, see [2, 21]. A main result of [24] asserts that the canonical central extension Ά = T*P © (P x IR) as described at the end of Sect. 2 is integrable (i.e. admits a global groupoid) provided that the Poisson manifold P admits a symplectic groupoid. Theorem 3.1 [24]. 1. If the symplectic groupoid (Γ z£ P, α 0 , j80) is a-simply connected, and prequantizable, then there exists a unique prequantίzation E off such that the identity space P has no holonomy. 2. Let £ Λ f be such a prequantization with connection θ so that E has no holonomy over εo(P), where ε0: P -> Γ is the unit map. Let ε: εo(P)-> E be any horizontal section over εo(P); then there exists a canonical groupoid structure on E over P, with unit map ε o ε 0 , source α = α 0 ° π, and target β = β0 o π, such that π: E -+ Γ is a groupoid homomorphism. 3. The Lie algebroid of (E zX P, α, β) is isomorphic to the canonical central extension i = Γ * P φ ( P x ! R ) introduced in Sect. 2. E is called the canonical central extension of the symplectic groupoid Γ.
4. Symplectic Realizations and Classical Intertwiner Spaces A symplectic realization (X, J) of a Poisson manifold P is defined as a symplectic manifold X together with a Poisson map J : X -> P which is complete in the sense that the pullback to Xof every compactly supported function on P has a complete Hamiltonian vector field. Symplectic realizations are generally considered as classical analogue of representations of an associative algebra. Morphisms between realizations are, by definition, lagrangian submanifolds of X2 x X\ (the bar denotes reversing the sign of the symplectic structure) contained in the coisotropic submanifold X2 * X\ = (Ji x J\) ~ * (diagonal in P x P). We however note that these do not form a category because of the usual clean intersection requirements for good compositions [27, 26].
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Now suppose that P is integrable as a Poisson manifold in the sense that it admits an α-simply connected symplectic groupoid Γ. Given such a symplectic groupoid (Γ zj P, α, β\ by a symplectic left Γ-module, we mean a symplectic manifold X together with a smooth map J: X -+ P such that X Λ P admits a symplectic left Γ-action (see [17, 27] for the definition of symplectic groupoid action). J is usually called the momentum map of the module X. The symplectic left modules of a given symplectic groupoid Γ become a "category" ζ(Γ) in the following sense: the objects of ζ(Γ) are symplectic left Γ-modules, the morphisms are canonical relations satisfying certain conditions compatible with the groupoid actions, and the composition of morphisms is the set-theoretic composition of relations. More precisely, a morphism from a symplectic left Γ-module X2 to a symplectic left Γ-module Xί is a lagrangian submanifold $£ c X2 * Xx that is invariant under the diagonal action of Γ. As usual, ζ(Γ) is not a true category and it requires the usual clean intersection assumption for the composition of morphisms being a morphism. It is shown [26] that symplectic realizations of a Poisson manifold are in fact in one-one correspondence with symplectic left-modules of its α-simply connected symplectic groupoid. For this reason, we will not distinguish these two concepts in the sequel. Theorem 4.1 [26]. Let Γ be an ot-sίmply connected symplectic groupoid over P. If X is a symplectic left Γ-module, then its momentum map p: X —• P is a symplectic realization. Conversely, any symplectic realization p: X -> P naturally admits a symplectic Γ-action so that it becomes a symplectic left Γ-module.
Let Jγ\ Xι-+ P and J2: %i ~+ P be any two symplectic realizations. For technical reasons, we always assume that, as symplectic left Γ-modules, Γ acts on Xί and X2 freely and properly, and that the map J2 x J\'. Xi x l i -* P x P is transversal to the diagonal ΔP c P x P so that Γ\(X2 *X\) is a smooth manifold, where X2 * Xγ denotes the irfverse image of the diagonal ΔP under the map J2xJ1. It can be proved that Γ\(X2 * Xx) is a symplectic manifold with the symplectic structure being naturally induced from that on X1 xX2 (see Prop. 2.1 [27] or Theorem 1 [10]). We shall call this space the classical ίntertwiner space between the realizations J i : X1-^P and J2: X2 -* P. Heuristically, as the Poisson manifold P is quantized to an associative algebra st, the realizations J1: X1 -• P and J2: X2^P are quantized to representations πγ and π2 of the algebra J / . Then, the symplectic space Γ\(X2*Xi) expects to become the intertwiner space of the representations πx and π 2 . According to the "symplectic creed," a classical "state" is a lagrangian submanifold of the symplectic manifold. The following proposition shows that classical "states" of the symplectic manifold Γ \(X2 * Xi) are indeed the same as morphisms of the realizations. Thus, it gives another rationale for our terminology. Proposition 4.1. There is a one-one correspondence between morphisms of realizations J i. X ί -> P, J 2: X 2 -+ P and lagrangian submanifolds of the classical intertwiner space Γ\(X2 * X\).
Proof Let S£ be any morphism of the realizations J1 and J2, i.e., a lagrangian submanifold of X2*Xχ. By Proposition 3.1 in [26], Jδf is invariant under the diagonal action of Γ. It is thus simple to see by dimension counting that Γ\£P is a lagrangian submanifold of Γ \ ( X 2 * ^ i ) Conversely, the preimage of any
Classical Intertwiner Space and Quantization
lagrangian submanifold of the quotient space Γ\(X2*Xi) a lagrangian submanifold sitting inside X2 * X\.
479
is easily seen to be D
For Poisson groups, classical intertwiner spaces can be expressed in a simpler term. Suppose that G is a complete simply-connected Poisson group with simply-connected dual G* [3, 13]. Let m: G* xG* -^> G* denote the group multiplication. Analogous to the tensor product of representations of a Hopf algebra, there is an operation among symplectic realizations of G*, called the "tensor product" [25], which is described as follows. Let Jt: X1-+ G* and J2: X2-+G* be symplectic realizations of G*. Then Jt x J2: Xγ x X2 -» G* x G* is a symplectic realization of G* x G*. Since the group multiplication m: G* xG* -+ G* is a Poisson map, X χ x X 2 m °( J i χ J *>>G* is thus a symplectic realization of G*. We write this realization as X x ® X 2 , and the realization map m°(Jγ x J2) as Jx ® J 2 . In this way, we obtain an associative product on the set of all realizations of G*5 which is called the "tensor product." The α-simply connected symplectic groupoid D of the Poisson group G* is diffeomorphic, as a differential manifold, to GxG*, and the groupoid structure is defined in terms of dressing actions [12]. Given a symplectic realization J: X -> G*5 there is a symplectic groupoid D-action on X as discussed above, as well as a Poisson group G-action as introduced by Lu [11] by considering J as the momentum mapping. It was shown [25] that these two actions are in fact equivalent in the sense that one action can be expressed naturally in terms of the other. In particular, the Poisson manifold resulting from the groupoid reduction D\X [27, 29, 25] is Poisson diffeomorphic to the Poisson group reduction X/G as defined by Semenov-Tian-Shansky [18]. The following proposition can also be easily verified. Proposition 4.2. Let J x : X1 -* G* and J2: X2 -> G* be symplectic realizations of G*. Then, the classical intertwiner space D\(X2 *X±) is symplectic diffeomorphic to the symplectic reduction (J2 (x) Jx)~ 1 (1 G *)/G {see [ l l ] / o r symplectic reduction in the context of Poisson group actions), where J2: X2 -> G* denotes the inverse realization defined by J 2 (x) = (J 2 (x))~ 1 , and 1G* is the unit ofG*. As a consequence, we conclude that to study classical intertwiner spaces for Poisson groups, it suffices to confine one's attention to the symplectic reduction J~1(IG*)/G for a certain symplectic realization J: X -+ G*. As an immediate consequence of Proposition 4.1, we have the following result generalizing Theorem 2.6 in [5]. Corollary 4.1. Let G be a simply connected Poisson Lie group with G* being its simply connected dual group. Let X be a symplectic manifold with a Poisson group G-action and J: X -» G* its corresponding momentum mapping [11]. There is a oneone correspondence between G-invariant lagrangian submanifolds of X and lagrangian submanifolds of the reduced space J~i(ίG*)/G, where 1G* is the unit of G*. 5. Prequantization of Symplectic Realizations This section can be considered as a continuation of Sect. 2, aimed at the global feature. Since we are mostly interested in the case of symplectic groupoids in the
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present paper, we shall concentrate on the particular example of the central extension arising from Poisson manifolds as described at the end of Sect. 2. Let P be a Poisson manifold with Poisson tensor π, and (Γ zχP,a0, β0) an α-simply connected symplectic groupoid. Let J: X -> P be a symplectic realization. Suppose that X is prequantizable and φ: L -• X is a prequantum circle-bundle of X with connection 0L. To describe our construction, we first proceed at the level of Lie algebroids. As in Sect. 2, let A —A © (P x 1R) be the canonical central extension of T*P corresponding to the Poisson manifold P. The connection ΘL thus induces a Lie algebroid action [7] of A on the prequantum circle-bundle L, as described by the following: Proposition 5.1. Let J = J°φ: L -+ P be the composition of φ: L -> X with J:X-+P. Then, (ω9f)eΓ{Ά)->Xj.ω-(J*f)ζeSΪ(L) defines a Lie algebroid action, where ωeΩί(P\feCco(P), Xj*ω is the horizontal lift of the vector field Xj*ω, and ζ is the Euler vector field on L generating the circle action. The proof is straightforward and is left to the reader. As in the case of Lie algebras, one expects to integrate a Lie algebroid action to a Lie groupoid action. This can be done under certain mild conditions. Below, we shall outline this result briefly. There are in fact two obstructions to such an integration: a dynamical one and a topological one. The dynamical obstruction is, roughly speaking, that certain vector fields involved need to be complete. More precisely, we say that a Lie algebroid action on a manifold M: Γ(^)->^(M), η->Xηe&(M), VηeΓ(A), is complete iϊXη is complete whenever η, considered as a left invariant vector field on Γ, is complete. The topological condition requires a certain assumption on simply-connectivity, which in our case, means that Γ is α-simply connected. Proposition 5.2. Let (Γ zj P, α, β) be an oc-connected and simply connected Lie groupoid with Lie algebroid A-> P. Then a Lie groupoid Γ-action on J: S -> P corresponds to a complete Lie algebroid action. Conversely, any complete Lie algebroid action on J: S -> P can be integrated to a Lie groupoid Γ-action. This proposition can be proved by imitating the proof of Theorem 3.1 in [26], which is left to the reader. We now turn to the particular case as in Proposition 5.1. The groupoid corresponding to A is the central extension groupoid (£ ij P, α, jS) as described in Theorem 3.1. Certainly, we expect that this Lie algebroid action extends to a groupoid £-action. However, one cannot simply apply Proposition 5.2, since the groupoid (E ij P, α, β) is generally not α-simply connected. In fact, we have the following Proposition 5.3. The central extension groupoid (E zt P, α, β) is ot-simply connected iff the cohomology class of Ω\a~^u) does not vanish in H2((XQ 1(U\ JR) for any ueP, where Q\a-i(u) denotes the restriction of the symplectic form Ω ofΓ on the fiber OCQ 1{U). Proof. For any ueP, α~1(w) is a principal Γ 1 -bundle over α^1(w), with Ω|α-i(M) being the curvature. The conclusion thus follows immediately by applying Gysin sequence [1]. •
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Below, we shall get around this difficulty using the method adopted from [24]. Our main result is the following: Theorem 5.1. The central extension groupoίd (E z% P, a, β\ naturally acts on J = J°φ: L -+ P, lifting the Lie algebroid action of A on L as defined in Proposition 5.1. Let 9 be the distribution on ExLxL defined by @ = {{aω (f)ξ,9 Xj*ω-(J*f)ζ)\ωeΩί(P)JeCGC(P)}. Here ξ and £, respectively, denote the vector fields on E and L generating their circle actions. Let / be the submanifold of ExLxL consisting of all elements of the form (J(/), /, /) for any leL, where P is considered as a submanifold of E via the embedding ε o ε0. / is clearly transversal to the distribution @. Let Ξ be the minimal ^-invariant submanifold containing / and immersed in ExLxL obtained by the method of characteristics. It suffices to show that Ξ is a graph over the submanifold E2 = {(κ, l)eEx L Iβ(κ) = J(l)}. Obviously, the image of Ξ under the natural projection p: ExLx L -> ExL given by p(κ91, m) = (κ, I), is E2. To prove the other half, we note that the two-torus T2 naturally acts on ExLxL by (s, t)'(κ91, m) = (S'K9tΊ, (sH-f) m), for any (TC, /, mjeExLxL
and s, teT1 .
Thus, (ExLxL )/T2 -^ΓxXxX with p([κ9 /, m]) = (π(4 φ(l\ φ(m)) is a_circle bundle, which is in fact a prequantization of Γ x X x X, with connection Θ. The latter is the one-form on (ExLxL )/T2 naturally induced from Θ = (θ,θL, — θL)e Ωι(ExLxL). Below, we shall characterize some basic properties of the manifold Ξ in the following series of lemmas. Lemma 5.1. The T2-action leaves Ξ invariant. Proof Since (ξ, 0, ζ)e9f9 then (κ91, m)eΞ iff (t κ, I, t m)eΞ for any teT1. For the action of the other generator of Γ 2 , we note that, by definition, (κ9 l,m)eΞ iff κ = Φ*(u)9 and m = ΦJ(l)9 where u = β(κ) = J(l)eP, Φα is a product of flows on E generated by vector fields of the form XΛ*ω-(aι*f)ξ for ωeΩ^P) and /eC°°(P), and Φ^is the corresponding product of flows on L, in the same order, generated by Xj*ω — (J*f)ζ- Since Xj*ω — (J*f)ζ commutes with the Euler vector field ζ for any J J 1 (ωJ)eΓ(Άl it follows that t-m = t Φ (l) = Φ (t-l) for any teT . Therefore, (κ9tΊ9t' m)eΞ iff (κ9 /, m)eΞ. Lemma 5.2. The pull back of Θ on Ξ is zero. Proof. We denote by λ the natural projection Ex LxL -> Γ xX xX, and write A=λ~ί(A), where A a Γ x X x X is the graph of the groupoid action. Let i be the natural inclusion from A to Ex LxL. It is easy to check that d(i*Θ) = 0 and i * Θ is nowhere zero in A. Hence, the equation i*ΓxXxX. (2) It is worth noting that the symplectic groupoid Γ itself does not act on the circle-bundle L in general. In fact, on the Lie algebroid level, whether an action exists depends on whether the Lie algebroid A = T*P can be embedded as a subalgebroid of the canonical central extension A = T*P © ( P x R ) . The latter is, however, related to the existence of a Liouville vector field on the Poisson manifold P, as pointed out by Lu (see the remark at the end of Sect. 2). 6. Prequantization of the Classical Intertwiner Space
The purpose of this section is to give an explicit construction of the prequantum bundle for the classical intertwiner space, using the prequantum bundles of the realizations and the symplectic groupoid. First, we need a couple of lemmas. The first one is a direct consequence of Theorem 5.1. Lemma 6.1. Under the same assumption as in Theorem 5.1, then (1) the projection λ = (π, φ, φ): ExLxL -^ Γ xXxX maps the graph of groupoid action Ξ a ExLxL to the graph of groupoid action A cz A x X x X. I.e.,/or any τeE
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483
and weL such that β(τ) = J{w\ we have
φ(τ w) = π(τ)*φ(w) (2) on L, the groupoid E-action commutes with the circle-action. Proof. To prove (1), let Q) be the distribution in ExLxL as defined in the previous section. It is simple to note that λ pushes down any vector in 2 to a vector tangent to the graph A. Also, it is trivial to see that λ sends / to a submanifold of A. The conclusion thus follows immediately from the construction of Ξ. For (2), we note that infinitesimally the groupoid E-action on L is generated by vector fields of the form Xj*ω-(J*f)ζ for ωeΩ^P) and/eC°°(P) (cf. Proposition 5.1). It is easy to check that any vector field of this form commutes with ζ. This concludes the proof of the lemma. • Lemma 6.2. Under the same assumption as in Theorem 5A,ΘL is invariant under the groupoid E-action. I.e., for any τeE such that β(τ) = u and oc(τ) = v, ΦΪ{ΘL\U) = ΘL\V,
where ΘL\U and ΘL\V are the restrictions of ΘL on the fibers φ~ι{u) and φ'1^) respectively, and Φτ: φ~1(u)-+φ~1(v) denotes the diffeomorphism of φ-fibers multiplying by the groupoid element τ. Proof For any (ω,f)eΓ(A\
we have
£;W(^AH*J*ω~μ*/)O J dθL + dt(Xj*ω-(J*f)ζ) J 0L] = {Xj.ω-(J*f)ζ) J φ*Ωx + d(-J*f) =
Xj*ωJφ*Ωx-d(J*f)
where Ωx is the symplectic form on X. It thus follows that Lχj,ω_{j*f)ζθL = 0, iff ω = df Therefore, the flow of the vector field Xj*df—(J*f)ζ preserves the connection form ΘL. Since any point τeE can be reached by a product of flows generated by such vector fields, it thus follows that * I « ) = ΘLL where u = β(τ) and v = φ). • Let Jx: Xx-+ P and J 2 Xi -+ P be any two symplectic realizations of P, on which the symplectic groupoid (Γ z£ P, α 0 , βo) acts freely and properly. Suppose that φ1\L1-^X1 and φ2: L2->Xi are prequantum bundles with connection forms θLχ and θLl, respectively. It follows from Lemma 6.1 that E\(L2*L1\ where the groupoid E acts on (L 2 * L i ) diagonally, is a circle bundle over Γ\(X2 *Xi). Theorem 6.1. The one-form ( — ΘL2, θLι) descends to a connection form on the circle bundle E\{Γ2*L1)^Γ\(1Γ2*X1)9 which is in fact a prequantum bundle. Proof According to Lemma 6.2, both θLι and — θLz are invariant under the groupoid £-action. To show that ( — θL2,θLl) descends to the quotient space £ \ ( L 2 *Li), it suffices to show that the restriction of ( —0L 2 , θLί) to any diagonal £-orbit is zero. Let Ji=Jι° φι and J2 = J2° φ2, and let d and ζ2 denote the Euler vector fields on Li and L 2 , respectively. We note that the tangent space to the diagonal £-orbit
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at any (z2, z^eJ^iiήxJϊ ^) is spanned by vectors of the form (2jfθ{z2)-f(u)ζ2, ^jf ^i)-/(w)Ciλ where θeΩ\P) and/eC°°(P). It is simple to see that (Xj?θ(z2)-f(u)ζ2,Xjrθ(z1)-f(u)ζ1)_A(-θL2,θLι)
=f(u)-f(u) = 0. Thus, ( — θLl, θLl) descends to a well defined one-form on the quotient space. It is simple to see that it is indeed a connection form defining a prequantum bundle. • As discussed in Sect. 4, if the Poisson manifold P is a Poisson group G*, any classical intertwiner space would reduce to the Poisson group reduction J~1(1G*)/G for a certain symplectic realization J:X-^G*. In this case, the prequantum bundle can also be described in a simpler form. Let C be the level set J~ 1 (1G*) Suppose that φ: L —• X is a prequantum bundle with connection ΘL. Then, the map ξeβ-+Xξ\ceSr(φ-1(C))
(2)
is indeed a Lie algebra homomorphism since the intersection of C with any G-orbit is isotropic. Here Xξ\c denotes the horizontal lift on φ~1(C) of the infinitesimal generator Xξ of the Lie algebra g-action. Therefore, Eq. (2) also defines a Lie group G-action on φ~x(C) if G is simply connected. However, we should note that, in general, it is impossible to lift the G-action to the entire bundle L. ι
Proposition 6.1. The pull back of the connection form ΘL on φ~ (C) descends to a one-form on the quotient space φ~1(C)/G, which defines a prequantum connection 1 1 on the circle bundle φ
Proof Let E A D be a prequantum bundle of the symplectic groupoid (D z£ G* , α 0 , β0), and l e £ a n y element over the point {1G} x {l G *}eD( = G x G*). Since the identity space G*, considered as a lagrangian submanifold of D (i.e., being identified with {1Q} X G *
